#7656580

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(5,6)(7,10)(8,11)(9,12)(14,19)(15,23)(21,22)(24,25);(1,2)(3,4)(14,19)(20,26)(21,22)(27,28);(5,7)(6,9)(8,12)(10,11)(13,14,18,19)(15,20,23,26)(16,21,17,22)(24,27,25,28);(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,27)(22,28);(5,7)(6,10)(8,12)(9,11)(13,16,18,17)(14,21,19,22)(15,24,23,25)(20,27,26,28);(5,6)(7,9)(8,11)(10,12)(13,17,18,16)(14,22,19,21)(15,25,23,24)(20,28,26,27)

Polycyclic presentation:

2^9|c^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7563776);